// Make newform 2850.2.a.k in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2850_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_2850_2_a_k();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_2850_2_a_k();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_2850_a();" function MakeCharacter_2850_a() N := 2850; order := 1; char_gens := [1901, 1027, 1351]; v := [1, 1, 1]; // chi(gens[i]) = zeta^v[i] assert UnitGenerators(DirichletGroup(N)) eq char_gens; F := CyclotomicField(order); chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]); return MinimalBaseRingCharacter(chi); end function; function MakeCharacter_2850_a_Hecke(Kf) return MakeCharacter_2850_a(); end function; function ExtendMultiplicatively(weight, aps, character) prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec primes := PrimesUpTo(prec); prime_powers := primes; assert #primes eq #aps; log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1) F := Universe(aps); FXY := PolynomialRing(F, 2); // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ... R := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 2; raw_aps := [[-1], [1], [0], [0], [-4], [4], [-6], [-1], [0], [-2], [0], [4], [-12], [-6], [0], [14], [-10], [-6], [4], [12], [-8], [-8], [-12], [-8], [10], [12], [-2], [-20], [-14], [10], [10], [-20], [6], [-12], [0], [16], [-6], [10], [-12], [2], [6], [-10], [6], [10], [10], [-4], [-8], [-10], [12], [-10], [-14], [10], [26], [-24], [2], [-16], [-2], [-28], [14], [12], [2], [-14], [0], [-18], [-4], [-18], [8], [-18], [-32], [34], [-22], [34], [12], [-12], [12], [-36], [-4], [-18], [28], [-34], [8], [-14], [-12], [18], [16], [12], [-32], [32], [24], [-16], [24], [38], [18], [36], [-4], [-16], [18], [-12], [0], [2], [-24], [30], [36], [32], [4], [8], [-32], [22], [-12], [-42], [-30], [46], [-6], [-28], [28], [36], [26], [24], [-18], [-26], [-2], [-26], [30], [12], [-4], [20], [22], [6], [40], [26], [20], [44], [-8], [2], [-30], [50], [38], [-48], [42], [26], [-40], [-4], [24], [-12], [50], [8], [50], [42], [-28], [24], [28], [42], [2], [0], [-12], [-48], [4], [-42], [20], [58], [-40], [-42], [44], [38], [-34], [-56], [48], [2], [-14], [-22], [10], [-2], [-14], [-44], [-56], [38], [28], [0], [-38], [-10], [-28], [-34], [8], [-10], [-64], [-24], [40], [-8], [-42], [-54], [-22], [-28], [8], [42], [32], [38], [18], [22], [54], [24], [42], [0], [-20], [-54], [-48], [34], [-12], [-68], [-6], [20], [4], [36], [4], [12], [-56], [-34], [-52], [-60], [32], [-54], [-70], [-40], [20], [-28], [28], [-74], [22], [-20], [-50], [-48], [-58], [-4], [-40], [0], [46], [-56], [-14], [-66], [-48], [-8], [44], [-28], [12], [-2], [62], [42], [24], [-10], [32], [56], [-46], [-30], [24], [22], [-14], [24], [-38], [32], [54], [-64], [-26], [12], [-2], [-16], [38], [-8], [-30], [-30], [48], [6], [-18], [-32], [40], [-56], [38], [-68], [32], [46], [70], [-36], [24], [40], [48], [10], [-70], [54], [-16], [22], [24], [-60], [16], [52], [-38], [-46], [16], [-4], [-64], [86], [-42], [26], [0], [6], [-56], [4], [-4], [-38], [-36], [-30], [-2], [-26], [-80], [-40], [-52], [8], [16], [10], [56], [36], [28], [6], [0], [40], [-2], [-66], [34], [18], [-80], [4], [-8], [54], [30], [-34], [40], [-36], [36], [4], [62], [22], [6], [16], [28], [50], [18], [-32], [-34], [18], [-34], [8], [-80], [-22], [4], [14], [-88], [-42], [-22], [-34], [38], [-68], [46], [-56], [90], [78], [-68], [10], [-38], [-74], [-50], [70], [-50], [-32], [24], [16], [58], [32], [14], [-42], [14], [48], [6], [-46], [14], [6], [76], [-54], [4], [12], [88], [-58], [-74], [60], [30], [6], [84], [26], [-92], [-26], [88], [-88], [-40], [100], [-14], [-82], [82], [6], [70], [-84], [-58], [48], [-92], [-80], [-78], [-22], [36], [-60], [8], [-84], [-12], [-46], [-82], [-16], [54], [-92], [-72], [-22], [26], [82], [-36], [-38], [20], [-42]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2850_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2850_2_a_k();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. function MakeNewformModFrm_2850_2_a_k(:prec:=1) chi := MakeCharacter_2850_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(2999) - 1; while true do trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]); B := Basis(S, prec + 1); S_basismat := Matrix([AbsEltseq(g): g in B]); if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then S_basismat := ChangeRing(S_basismat,Kf); f_lincom := Solution(S_basismat,trunc_vec); f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]]; return f; end if; error if prec eq maxprec, "Unable to distinguish newform within newspace"; prec := Min(Ceiling(1.25 * prec), maxprec); end while; end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_2850_2_a_k();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function MakeNewformModSym_2850_2_a_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2850_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![0, 1]>,<11,R![4, 1]>,<13,R![-4, 1]>,<23,R![0, 1]>],Snew); return Vf; end function;